higher frequency for the y-directional dispersion curve. Here, it can be clearly observed that the resonance frequencies are the same for both evaluations (x- and y-directions). This complements the analytic estimation that the two resonators are shared, meaning that two different eigen-modes occur at the same resonant frequency. Also, the impedance and the transmission curve resemble the ones corresponding to the Keff-substructure that were previously discussed in Chapter 3, Fig. 3.10(a). Here, it must be emphasized that the impedance matching condition (total transmission condition) is what we intend to employ for the total transmission superlensing phenomena.
In Fig. 5.5(b), the retrieved effective properties are presented. As estimated before, only the Kmy eff, term is involved in the resonant state while the My effm, term remains relatively constant. Unlike other works regarding metamaterials, the extreme stiffness value just above the resonant frequency is what we facilitate for the total transmission phenomena. All these exotic results originate from our uniquely designed elastic unit cell with local resonators connected by inclined slender beams.
5.4 Physical Explanation on Super-Resolution Imaging by
The solid line dispersion curve is numerically calculated by finite element analysis with commercial software COMSOL Multiphysics. The circles (for the x-direction) and squares (for the y-direction) are obtained by substituting calculated effective material parameters from Figs. 5.4 and 5.5 into the following basic periodic mass- spring dispersion relation in Eq. (3.23). Dispersion curves retrieved from both methods show a very good agreement. Note that the ranges of the resonant bandgaps in the Г-Χ and Г-Υ directions differ because they rely on different constitutive parameters. Specifically, only the stiffness term experiences a resonant state whereas the mass density term is almost constant in the Г-Υ (i.e., y) direction.
Exactly the opposite phenomenon occurs in the Г-Χ (i.e., x) direction. To realize total transmission in the y direction, we elaborately utilize the fact that the effective stiffness based resonant bandgap is formed below the resonant frequency
( 32 kHz from the analytical method) whereas the effective mass density based bandgap is formed above the resonant frequency. Therefore, the target frequency for total transmission in the Г-Υ direction is chosen to be 35.48 kHz which is slightly higher than the resonant frequency
. In this case, a pass band exists only for the Г-Υ direction while no wave can propagate in the Г-Χ direction because of the resonant bandgap.In Fig. 5.6(b), we plot the equi-frequency color contours for a range of frequencies around 35.48 kHz. Hyperbolic and nearly flat EFCs whose widths are several times larger than that of background medium can be clearly seen. It is also observable
that such dispersion behavior is valid for a wide range of frequencies. Similar to the canalization mechanism [113-115, 118], we can expect our slab-like elastic metamaterial lens to work as a transmission device that allows transportation of high wavevector components from one side to the other. With its hyperbolic dispersion, it enables an unbounded range of wave numbers to be delivered across the lens, ensuring no loss of any imaging information. Their EFCs that stretch out larger than that of a background medium make evanescent waves converted into propagating modes inside the metamaterial and transferred to the other side of the lens, thus preserving subwavelength information.
In fact, although not shown here, unit cells with similar geometries with different parameters were found to possess the desired characteristics, meaning that the unit cell microstructure in Fig. 5.2(a) is not the only solution. Although a small discrepancy exists in wave characteristics between the continuum body and the discrete mass-spring model, the analysis based on the discrete model provides adequate guidelines to analyze and obtain the desired parameters.
5.5 1-D Total Transmission Condition due to Impedance Matching Condition
In this section, we will show how the total transmission can be realized by using the designed metamaterial lens at the target operating frequency. Fig. 5.5(a), displays how the relative impedance Zy eff, Z varies as a function of frequency
along the Г-Υ direction. In the negative-stiffness bandgap region below the resonant frequency (
), the relative impedance has non-real values, meaning that no energy propagation is possible in that frequency range. Just above the resonant frequency, the real part of the impedance increases significantly as a result of the stiffness resonant state. The anomalously increased extreme stiffness term can thus counterbalance the reduced effective property (My effm, ) of the metamaterial lens.Slightly above the resonant frequency, the relative impedance is extremely large but it decreases as the frequency further increases. A perfectly-matched impedance (Zy eff, Z1) occurs at 35.48 kHz.
The one-dimensional transmission coefficient T for waves passing through a dissimilar medium can be formulated as
, ,
2
2cos( y ) y eff sin( y )
y eff
T Z Z
Nq l i Nq l
Z Z
, (5.19)
where N is the number of the unit cells embedded in the aluminum medium.
Therefore, Nl represents the total thickness of the metamaterial lens inserted in an aluminum medium.
Further verification of the total transmission is conducted with a continuum model and the simulation set-up is shown in Fig. 5.7. A periodic boundary condition is applied to upper and lower sides. Longitudinal plane waves incident along the x direction are excited by harmonic forces along the line source at the left side of the metamaterial layer. The transmitted wave profile is picked up at the right side.
PMLs are added at the left and right end sides to eliminate any reflected waves from boundaries.
The obtained transmission spectra based on Eq. (5.19) are also presented in the figure with a different number of unit cells (i.e., 1 and 10 unit cells). The transmission curve indicates that total transmission occurs at 35.48 kHz in the 10 unit cells case. The peak frequency makes good agreement with the impedance matching frequency for the single unit cell case. Other observations can be made on additional peaks that occur from the Fabry-Perot resonances. As the number of unit cells increases, or when the lens becomes thicker, additional peaks appear owing to more frequency values satisfying the Fabry-Perot resonance conditions
y π
Nq ln (n is an integer). Correspondingly, more standing waves can be compressed within the thickness of the lens.
To better understand the total-transmission subwavelength-imaging mechanism, some analysis will be useful. When Zy eff, Z1, condition (5.19) becomes
1/ [cos( y) sin( y )] iNq ly
T Nq l i Nq l e and the magnitude of T becomes unity regardless of qy. This implies that any propagating wave in the y-direction can undergo total transmission. In other words, the above transmission relation is valid over an unlimited range of high transverse wavenumber components, meaning that they all can be converted to propagating waves and be transferred with total transmission.
It should be remarked that as in the zero-mass effect in an acoustic regime, the
realized total transmission in the elastic regime does not involve the Fabry-Perot resonance which is inappropriate for pulse type ultrasonic inspection.